Boolean Models Research Articles

An investigation on the robustness and accuracy of permeability estimation from 3D μ-CT images of rock samples based on the solution of Laplace's equation for pressure

This study aims to present a simple method to calculate the permeability of porous materials using μ-CT images and an approximation based on the solution of Laplace's equation for pressure. For this purpose, an in-house computer program was developed based on finite difference method to determine the distribution of pressure in voxelated pore space. Afterwards, approximate permeability was obtained using Euclidean distance map of the pore space and a simple upscaling scheme for a range of porous media including idealized channels of elementary cross sections, Boolean models of spherical grains, bundles of capillary tubes and digital rocks obtained from μ-CT imaging. Next, a comparison was made between the Laplace permeability approximation and the analytical solutions of idealized microstructure and digitally-computed permeability of a number of rock samples using the Stokes solver and the lattice-Boltzmann method. It has been revealed that the estimated permeability values are in good agreement over the investigated range of porosity. The developed Laplace permeability solver was also well matched with the results of experimental measurement on a real rock sample. As an illustration of the applicability of the proposed method, the anisotropy of permeability at pore scale and the cross-correlation between permeability and connected macro-porosity was analyzed for the experimentally investigated rock sample. In conclusion, the results of this paper suggest that the proposed permeability solver is suitable for rough estimation of permeability and hence rough evaluation of heterogeneity/anisotropy from μ-CT images of rocks, particularly for large datasets with high number of pore voxels.

The Tractability of SHAP-Score-Based Explanations for Classification over Deterministic and Decomposable Boolean Circuits

Scores based on Shapley values are widely used for providing explanations to classification results over machine learning models. A prime example of this is the influential SHAP-score, a version of the Shapley value that can help explain the result of a learned model on a specific entity by assigning a score to every feature. While in general computing Shapley values is a computationally intractable problem, it has recently been claimed that the SHAP-score can be computed in polynomial time over the class of decision trees. In this paper, we provide a proof of a stronger result over Boolean models: the SHAP-score can be computed in polynomial time over deterministic and decomposable Boolean circuits. Such circuits, also known as tractable Boolean circuits, generalize a wide range of Boolean circuits and binary decision diagrams classes, including binary decision trees, Ordered Binary Decision Diagrams (OBDDs) and Free Binary Decision Diagrams (FBDDs). We also establish the computational limits of the notion of SHAP-score by observing that, under a mild condition, computing it over a class of Boolean models is always polynomially as hard as the model counting problem for that class. This implies that both determinism and decomposability are essential properties for the circuits that we consider, as removing one or the other renders the problem of computing the SHAP-score intractable (namely, #P-hard).

Left–right crossings in the Miller–Abrahams random resistor network and in generalized Boolean models

We consider random graphs G built on a homogeneous Poisson point process on Rd, d≥2, with points x marked by i.i.d. random variables Ex. Fixed a symmetric function h(⋅,⋅), the vertexes of G are given by points of the Poisson point process, while the edges are given by pairs {x,y} with x≠y and |x−y|≤h(Ex,Ey). We call GPoissonh-generalized Boolean model, as one recovers the standard Poisson Boolean model by taking h(a,b)≔a+b and Ex≥0. Under general conditions, we show that in the supercritical phase the maximal number of vertex-disjoint left–right crossings in a box of size n is lower bounded by Cnd−1 apart from an event of exponentially small probability. As special applications, when the marks are non-negative, we consider the Poisson Boolean model and its generalization to h(a,b)=(a+b)γ with γ>0, the weight-dependent random connection models with max-kernel and with min-kernel and the graph obtained from the Miller–Abrahams random resistor network in which only filaments with conductivity lower bounded by a fixed positive constant are kept.

Lac Operon Boolean Models: Dynamical Robustness and Alternative Improvements

In Veliz-Cuba and Stigler 2011, Boolean models were proposed for the lac operon in Escherichia coli capable of reproducing the operon being OFF, ON and bistable for three (low, medium and high) and two (low and high) parameters, representing the concentration ranges of lactose and glucose, respectively. Of these 6 possible combinations of parameters, 5 produce results that match with the biological experiments of Ozbudak et al., 2004. In the remaining one, the models predict the operon being OFF while biological experiments show a bistable behavior. In this paper, we first explore the robustness of two such models in the sense of how much its attractors change against any deterministic update schedule. We prove mathematically that, in cases where there is no bistability, all the dynamics in both models lack limit cycles while, when bistability appears, one model presents 30% of its dynamics with limit cycles while the other only 23%. Secondly, we propose two alternative improvements consisting of biologically supported modifications; one in which both models match with Ozbudak et al., 2004 in all 6 combinations of parameters and, the other one, where we increase the number of parameters to 9, matching in all these cases with the biological experiments of Ozbudak et al., 2004.

Integrating Patient-Specific Information into Logic Models of Complex Diseases: Application to Acute Myeloid Leukemia.

High throughput technologies such as deep sequencing and proteomics are increasingly becoming mainstream in clinical practice and support diagnosis and patient stratification. Developing computational models that recapitulate cell physiology and its perturbations in disease is a required step to help with the interpretation of results of high content experiments and to devise personalized treatments. As complete cell-models are difficult to achieve, given limited experimental information and insurmountable computational problems, approximate approaches should be considered. We present here a general approach to modeling complex diseases by embedding patient-specific genomics data into actionable logic models that take into account prior knowledge. We apply the strategy to acute myeloid leukemia (AML) and assemble a network of logical relationships linking most of the genes that are found frequently mutated in AML patients. We derive Boolean models from this network and we show that by priming the model with genomic data we can infer relevant patient-specific clinical features. Here we propose that the integration of literature-derived causal networks with patient-specific data should be explored to help bedside decisions.

Logical matrix factorization towards topological structure and stability of probabilistic Boolean networks

Abstract The study of logical matrix factorization provides a new insight into the matrix dimension reduction problems of biological systems. This paper develops the logical matrix factorization technique for exploring the topological structure and stability of probabilistic Boolean networks (PBNs). Firstly, the union set of distinct indices in factorized structural matrices for different modes is obtained, based on which, a size-reduced system is constructed for the original PBN. Secondly, it is proved that the topological structure of original PBN is equivalent to that of the size-reduced system. Thirdly, the equivalence of finite-time stability and stability in distribution between the original PBN and the size-reduced system is further revealed. Finally, the effectiveness of the obtained new results are verified via several Boolean models of genetic regulatory networks (GRNs).

Asymptotic behavior of mean density estimators based on a single observation: the Boolean model case

The mean density estimation of a random closed set in $$\mathbb {R}^d$$ , based on a single observation, is a crucial problem in several application areas. In the case of stationary random sets, a common practice to estimate the mean density is to take the n-dimensional volume fraction with observation window as large as possible. In the present paper, we provide large and moderate deviation results for these estimators when the random closed set $$\Theta _n$$ belongs to the quite general class of stationary Boolean models with Hausdorff dimension $$n<d$$ . Moreover, we establish a central limit theorem and a Berry–Esseen bound for the family of estimators under study. Our findings allow to recover some well-known results in the literature on Boolean models. Finally, we also provide a guideline for the estimation of the mean density of non-stationary Boolean models characterized by high intensity of the underlying Poisson point process.

Reduction and Analysis of Boolean Control Networks by Bisimulation

The concept of bisimulation has been recently introduced in the study of Boolean control networks (BCNs). The notion provides a perspective that explains how a state trajectory of one BCN can be paired with a state trajectory of another BCN, and vice versa. The present paper further pursues the notion of bisimulation in BCNs. It first gives necessary and sufficient conditions for checking this notion. The conditions are more general and neater than those previously published; more importantly, they allow us to go further to consider reducing a BCN via bisimulation. A bisimulation-based methodology is then proposed to reduce the size of the state space of a BCN. Applying the reduction methodology allows one to infer control properties of a BCN from those of the corresponding reduced system. Possibilities of applications of the results are illustrated on several Boolean models of real biomolecular systems.

FPGA Accelerated Analysis of Boolean Gene Regulatory Networks.

Boolean models are a powerful abstraction for qualitative modeling of gene regulatory networks. With the recent availability of advanced high-throughput technologies, Boolean models have increasingly grown in size and complexity, posing a challenge for existing software simulation tools that have not scaled at the same speed. Field Programmable Gate Arrays (FPGAs) are powerful reconfigurable integrated circuits that can offer massive performance improvements. Due to their highly parallel nature, FPGAs are well suited to simulate complex molecular networks. We present here a new simulation framework for Boolean models, which first converts the model to Verilog, a standardized hardware description language, and then connects it to an execution core that runs on an FPGA coherently attached to a POWER8 processor. We report an order of magnitude speedup over a multi-threaded software simulation tool running on the same processor on a selection of Boolean models. Analysis on a T-cell large granular lymphocyte leukemia (T-LGL) demonstrates that our framework achieves consistent performance improvements resulting in new biological insights. In addition, we show that our solution allows to perform attractor detection at an unprecedented speed, exhibiting a speedup ranging from one to three orders of magnitude compared to alternative software solutions.

A survey of gene regulatory networks modelling methods: from differential equations, to Boolean and qualitative bioinspired models

Gene Regulatory Networks (GRNs) represent the interactions among genes regulating the activation of specific cell functionalities, such as reception of (chemical) signals or reaction to environmental changes. Studying and understanding these processes is crucial: they are the fundamental mechanism at the basis of cell functioning, and many diseases are based on perturbations or malfunctioning of some gene regulation activities. In this paper, we provide an overview on computational approaches to GRN modelling and analysis. We start from the biological and quantitative modelling background notions, recalling differential equations and the Gillespie’s algorithm. Then, we describe more in depth qualitative approaches such as Boolean networks and some computer science formalisms, including Petri nets, P systems and reaction systems. Our aim is to introduce the reader to the problem of GRN modelling and to guide her/him along the path that goes from classical quantitative methods, through qualitative methods based on Boolean network, up to some of the most relevant qualitative computational methods to understand the advantages and limitations of the different approaches.

Antifragility Predicts the Robustness and Evolvability of Biological Networks through Multi-Class Classification with a Convolutional Neural Network.

Robustness and evolvability are essential properties to the evolution of biological networks. To determine if a biological network is robust and/or evolvable, it is required to compare its functions before and after mutations. However, this sometimes takes a high computational cost as the network size grows. Here, we develop a predictive method to estimate the robustness and evolvability of biological networks without an explicit comparison of functions. We measure antifragility in Boolean network models of biological systems and use this as the predictor. Antifragility occurs when a system benefits from external perturbations. By means of the differences of antifragility between the original and mutated biological networks, we train a convolutional neural network (CNN) and test it to classify the properties of robustness and evolvability. We found that our CNN model successfully classified the properties. Thus, we conclude that our antifragility measure can be used as a predictor of the robustness and evolvability of biological networks.

The Case for Algebraic Biology: from Research to Education.

Though it goes without saying that linear algebra is fundamental to mathematical biology, polynomial algebra is less visible. In this article, we will give a brief tour of four diverse biological problems where multivariate polynomials play a central role-a subfield that is sometimes called algebraic biology. Namely, these topics include biochemical reaction networks, Boolean models of gene regulatory networks, algebraic statistics and genomics, and place fields in neuroscience. After that, we will summarize the history of discrete and algebraic structures in mathematical biology, from their early appearances in the late 1960s to the current day. Finally, we will discuss the role of algebraic biology in the modern classroom and curriculum, including resources in the literature and relevant software. Our goal is to make this article widely accessible, reaching the mathematical biologist who knows no algebra, the algebraist who knows no biology, and especially the interested student who is curious about the synergy between these two seemingly unrelated fields.

Personalizing the Top-k Spatial Keyword Preference Query with textual classifiers

The volume of data available online has increased considerably in the last years. Among this extensive amount, there is a specific kind called spatial data – a representation of a physical object using its spatial coordinates (e.g. latitude and longitude). Spatial queries are widely employed to manipulate spatial data more efficiently. However, the user has a crucial role in the spatial information retrieval process when querying the needed information. During the last few decades, researchers have proposed several techniques to aid users to express the information needed, such as boolean models, pattern matching operators, and query expansion. Despite the existence of significant alternatives to express the information need, there is still a lack of solutions applied to keyword preference queries. In this context, this work proposes a personalization approach to improve the Top-k Spatial Keyword Preference Query (SKPQ) results. By exploiting reviews on points of interest, the system identifies points which satisfy the user to order the result set with respect to his/her preference. The SKPQ result is compared to the one generated by our proposal using the OpinRank dataset. The assessment indicates a relative NDCG (Normalized Discounted Cumulative Gain) improvement of the proposed approach over the SKPQ of 92% when using random query keywords, and 33% when using frequent keywords. This personalization approach can be applied to any spatial keyword query since it re-orders the results instead of modifying the query processing.

Exact solving and sensitivity analysis of stochastic continuous time Boolean models

BackgroundSolutions to stochastic Boolean models are usually estimated by Monte Carlo simulations, but as the state space of these models can be enormous, there is an inherent uncertainty about the accuracy of Monte Carlo estimates and whether simulations have reached all attractors. Moreover, these models have timescale parameters (transition rates) that the probability values of stationary solutions depend on in complex ways, raising the necessity of parameter sensitivity analysis. We address these two issues by an exact calculation method for this class of models.ResultsWe show that the stationary probability values of the attractors of stochastic (asynchronous) continuous time Boolean models can be exactly calculated. The calculation does not require Monte Carlo simulations, instead it uses graph theoretical and matrix calculation methods previously applied in the context of chemical kinetics. In this version of the asynchronous updating framework the states of a logical model define a continuous time Markov chain and for a given initial condition the stationary solution is fully defined by the right and left nullspace of the master equation’s kinetic matrix. We use topological sorting of the state transition graph and the dependencies between the nullspaces and the kinetic matrix to derive the stationary solution without simulations. We apply this calculation to several published Boolean models to analyze the under-explored question of the effect of transition rates on the stationary solutions and show they can be sensitive to parameter changes. The analysis distinguishes processes robust or, alternatively, sensitive to parameter values, providing both methodological and biological insights.ConclusionUp to an intermediate size (the biggest model analyzed is 23 nodes) stochastic Boolean models can be efficiently solved by an exact matrix method, without using Monte Carlo simulations. Sensitivity analysis with respect to the model’s timescale parameters often reveals a small subset of all parameters that primarily determine the stationary probability of attractor states.

HETEROGENEITY ASSESSMENT BASED ON AVERAGE VARIATIONS OF MORPHOLOGICAL TORTUOSITY FOR COMPLEX POROUS STRUCTURES CHARACTERIZATION

Morphological characterization of porous media is of paramount interest, mainly due to the connections between their physicochemical properties and their porous microstructure geometry. Heterogeneity can be seen as a geometric characteristic of porous microstructures. In this paper, two novel topological descriptors are proposed, based on the M-tortuosity formalism. Using the concept of geometric tortuosity or morphological tortuosity, a first operator is defined, the H-tortuosity. It estimates the average variations of the morphological tortuosity as a function of the scale, based on Monte Carlo method and assessing the heterogeneity of porous networks. The second descriptor is an extension, named the H-tortuosity-by-iterativeerosions, taking into account different percolating particle sizes. These two topological operators are applied on Cox multi-scale Boolean models, to validate their behaviors and to highlight their discriminative power.

Current trends in information retrieval systems: review of fuzzy set theory and fuzzy Boolean retrieval models

This paper reviews the concept and goal of Information Retrieval Systems (IRSs). It also explains the synonymous concepts in Information Retrieval (IR)which include such terms as: imprecision, vagueness, uncertainty, and inconsistency. Current trends in IRSs are discussed. Fuzzy Set Theory, Fuzzy Retrieval Modelsare reviewed. The paper also discusses extensions of Fuzzy Boolean Retrieval Models including Fuzzy techniques for documents’ indexingandFlexible query languages. Fuzzy associative mechanisms were identified to include:(1)fuzzy pseudothesauri and fuzzy ontologies which can be used to contextualize the search by expanding the set of index terms of documents;(2)an alternative use of fuzzy pseudothesarui and fuzzy ontologies is to expand the query with related terms by taking into account their varying importance of an additional termand (3)fuzzy clustering techniques, where each document can be placed within several clusters with a given strength of belonging to each cluster, can be used to expand the set of the documents retrieved in response to a query.The paper concludesby recommending that in an electronic library environment, the librarians and information scientists should acquaint themselves with these terms in order to be more equipped in helping library users retrieve online documents relevant to their information needs.

Automated inference of Boolean models from molecular interaction maps using CaSQ

MotivationMolecular interaction maps have emerged as a meaningful way of representing biological mechanisms in a comprehensive and systematic manner. However, their static nature provides limited insights to the emerging behaviour of the described biological system under different conditions. Computational modelling provides the means to study dynamic properties through in silico simulations and perturbations. We aim to bridge the gap between static and dynamic representations of biological systems with CaSQ, a software tool that infers Boolean rules based on the topology and semantics of molecular interaction maps built with CellDesigner.ResultsWe developed CaSQ by defining conversion rules and logical formulas for inferred Boolean models according to the topology and the annotations of the starting molecular interaction maps. We used CaSQ to produce executable files of existing molecular maps that differ in size, complexity and the use of Systems Biology Graphical Notation (SBGN) standards. We also compared, where possible, the manually built logical models corresponding to a molecular map to the ones inferred by CaSQ. The tool is able to process large and complex maps built with CellDesigner (either following SBGN standards or not) and produce Boolean models in a standard output format, Systems Biology Marked Up Language-qualitative (SBML-qual), that can be further analyzed using popular modelling tools. References, annotations and layout of the CellDesigner molecular map are retained in the obtained model, facilitating interoperability and model reusability.Availability and implementationThe present tool is available online: https://lifeware.inria.fr/∼soliman/post/casq/ and distributed as a Python package under the GNU GPLv3 license. The code can be accessed here: https://gitlab.inria.fr/soliman/casq.Supplementary information Supplementary data are available at Bioinformatics online.

Integration of multisource data to support the identification of lateritic regolith in Eastern - Bahia, northeastern Brazil

This work used multi-source data integration techniques (gamma-spectrometry, magnetometry, SRTM-Altimetry) to identify areas favorable for the occurrence of well-developed lateritic regolith in the eastern region of the state of Bahia. The variables observed to define target potential were the high eTh/K and eU/K ratios and altitudes (obtained from SRTM). Based on these parameters, Boolean and fuzzy logic were applied to produce favorability maps. The best results were obtained under the fuzzy model (γ= 0.7) with a hit accuracy in the areas for potential laterite occurrence of 97.4% and a kappa value of 0.5, consisting of 118 control points obtained through fieldwork. The cross-referencing of the fuzzy image (γ = 0.7) with the magnetometry, total gradient (ASA-Analytical Signal Amplitude) suggested the predominance of more felsic protoliths in the most favorable areas. The lateritic index [IL = (eTh*eU)/K2)] was also applied, demonstrating good correlation with the areas determined by the Boolean and Fuzzy models. Mineralogical associations (clay minerals and/or iron oxides) of the target areas were estimated by the Crosta Technique in OLI/Landsat-8 images. The processing results were integrated in GIS environment, together with control points and data found in the bibliography (e.g. geological map, vertical electrical profile, drill-holes, occurrence of Fe and Al, lateritic crusts, geochemical anomalies).The observed correlation between the generated models and the direct data validates the effectiveness of the techniques used.

Cell phenotypes as macrostates of the GRN dynamics.

The two most fundamental processes describing change in biology, development, and evolution, occur over drastically different timescales. Development involves temporal sequences of cell states controlled by hierarchies of regulatory structures. It occurs over the lifetime of a single individual and is associated with gene expression level change of a given genotype. Evolution, by contrast, entails genotypic change through mutation, the acquisition/loss of genes and changes in the network topology of interactions among genes. It involves the emergence of new, environmentally selected phenotypes over the lifetimes of many individuals. We start by reviewing the most limiting aspects of the theoretical modeling of gene regulatory networks (GRNs) which prevent the study of both timescales in a common, mathematical language. We then consider the simple framework of Boolean network models of GRNs and point out its inadequacy to include evolutionary processes. As opposed to one-to-one maps to specific attractors, we adopt a many-to-one map which makes each phenotype correspond to multiple attractors. This definition no longer requires a fixed size for the genotype and opens the possibility for modeling the phenotypic change of a genotype, which is itself changing over evolutionary timescales. At the same time, we show that this generalized framework does not interfere with established numerical techniques for the identification of the kernel of controlling nodes responsible for cell differentiation through external signals.

Using activity time windows and logical representation to reduce the complexity of biological network models: G1/S checkpoint pathway with DNA damage

Biological systems are difficult to understand complex systems. Scientists continue to create models to simulate biological systems but these models are complex too; for this reason, new reduction methods to simplify complex biological models into simpler ones are increasingly needed. In this paper, we present a way of reducing complex quantitative (continuous) models into logical models based on time windows of system activity and logical (Boolean) models. Time windows were used to define slow and fast activity areas. We use the proposed approach to reduce a continuous ODE model into a logical model describing the G1/S checkpoint with and without DNA damage as a case study. We show that the temporal unfolding of this signalling system can be broken down into three time windows where only two display high level of activity and the other shows little or no activity. The two active windows represent a cell committing to cell cycle and making the G1/S transition, respectively, the two most important high level functions of cell cycle in the G1 phase. Therefore, we developed two models to represent these time windows to reduce time complexity and used Boolean approach to reduce interaction complexity in the ODE model in the respective time windows. The developed reduced models correctly produced the commitment to cell cycle and G1/S transfer through the expected behavior of signalling molecules involved in these processes. As most biological models have a large number of fast reactions and a relatively smaller number of slow reactions, we believe that the proposed approach could be suitable for representing many, if not all biological signalling networks. The approach presented in this study greatly helps in simplifying complex continuous models (ODE models) into simpler models. Moreover, it will also assist scientists build models concentrating on understanding and representing system behavior rather than setting values for a large number of kinetic parameters.